Pre-Quantale And The Categories Of Dual Quantale

The concept of quantale was introduced by C. J. Mulvey in 1986 with the purpose of studying the spectrum of C~*-algebra, as well as constructive foundations for quantum mechanics. The research of these subjects has related to several research area such as non-commutative C~*-algebra, the ideal theory of rings, linear logic and theoretic computer science. There are abundant contents in the structure of quantales, because quantale can be regard as the generalization of frame. This paper analyses and studied the relations among co-quantale, ideal and quantale congruence. We study carefully and deeply some properties of pre-quantale and the category of dual quantales. The arrangement of this paper is as follows:Chapter One Preliminary knowledge. In this chapter, we give the basic concepts and results of the theory of quantale and that of category which are used in the whole paper.Chapter Two The properties of co-quantale. In first part of this chapter, we study the ideal of co-quantale, and give the concrete construction of the ideal generated by an arbitary subset. Furthermore we study the relation between the ideal and the map, and obtain the equivalent condition that a map is an ideal nucleus. In the second part, we define a coquantale morphism and then prove the properties of these special elements don't change on the left adjoint of a morphism. Meanwhile, It's founded out the relationship betweencoquantale morphism the operation of a coquantale and obtainsthat an opened map of coquantales is a necessary sufficient condition for a coquantale morphism. In the thrird part we study some properities of idempotent quantale.Chapter Three Pre-conucleus and the relations of ideal and cogruence. In the final chapter, we define pre-conucleus and discuss some properties on it. Finally, in the view of quantale, the relation of ideal and cogruence is given. Moreover, the relation of their one to one is found by us.Chapter Four The category of dual quantales. In this part, we study mainly some special objects in the category of dual quantales. For example, initial object and terminal object, etc. According to these, we know that the category of dual quantales is not a pointed category. Then, the structure of equalizer is given. Therefore, we can prove that this category has the product. After that, we construct the limit and the inverse limit of the inverse system in this category.